Superelliptic Area Of $x^5+y^5=r^5$

Method due to Dirichlet, for your $r=1$ the area in the first quadrant is $$ \frac{\Gamma \left( \frac{6}{5} \right)^2}{\Gamma \left( \frac{7}{5} \right) } \approx 0.95015. $$

Since your number is larger than $1,$ i guess you are getting area for the whole thing, along the line $x+y = 0$ included. I suggest trying ratios of Gamma functions, see if you can get your $1.4017$